It seems to be an oddly US-centric aspect of how Americans see/idolise the president that you don't really get elsewhere. Its almost like criticising their father or similar.
Even something contentious like Brexit in the UK has less of a knee-jerk support for the leaders (e.g. many pro-Brexit commentards would not be so outraged by others pointing out the current PM is an uncaring cockwomble, for example, but would defend their political goal).
Even a symbolically powerful role like the monarch that also divides opinion fails to ignite the same pro-Trump/anti-Obama frothing as most UK 'republicans' may be against the idea of the monarchy but don't feel need to launch verbal rants against Liz herself.